First let me explain what the **index** in question refers to. If the math gets too full of jargon let me know in the comments.

In physics we are often interested in the spectrum of various operators on some manifolds we care about. Eg: the Dirac operator in 3+1 spacetime. In particular the low-energy long distance physics is contained in the zero modes (ground states).

Now what the "index" measures, for the Dirac operator $D$ and a given manifold $M$, is the difference between the number of left-handed zero modes and the number of right-handed zero modes. More technically:

$$ ind\,D = dim\,ker\,D - dim\,ker\,D^{+} $$

where $D$ is the operator in question; $ker\,D$ is the kernel of $D$ - the set of states which are annihilated by $D$; and $ker\,D^{+}$ is the kernel of its adjoint. Then, as you can see, $ind\,D$ counts the difference between the dimensionalities of these two spaces. This number depends only on the topology of $M$.

In short, the ASI theorem relates the topology of a manifold $M$ to the zero modes or ground states of a differential operator $D$ acting on $M$. This is obviously information of relevance to physicists.

Perhaps someone else can elaborate more on the physical aspects.

The best reference for this and other mathematical physics topics, in my opinion, is Nakahara.

This post imported from StackExchange Physics at 2014-04-01 16:37 (UCT), posted by SE-user user346